Nonlocal symmetries and ghosts.

*(English)*Zbl 1072.35193
Shabat, A.B.(ed.) et al., New trends in integrability and partial solvability. Proceedings of the NATO Advanced Research Workshop, Cadiz, Spain, June 12–16, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1835-5/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 132, 199-215 (2004).

The reviewed article has a problem character, because the theory of nonlocal symmetries of nonlocal differential equations remains incomplete. The reason is the absence of calculus for nonlocal vector fields. The author proposes a new theoretical and computational basis for nonlocal theory – nonlocal differential algebra, applicable to a wide variety of nonlocal systems, and suggests the concept of a ghost symmetry that resolves the Jacobi paradox of the failure of the Jacobi identity for nonlocal vector fields. The following topics of investigation are now of importance: (a) A complete reevaluation of earlier works on nonlocal symmetries of local and nonlocal partial differential equations, (b) The establishment of a complete nonlocal variational calculus on the nonlocal differential algebra \(\mathfrak{L}^{(\infty)}\), including nonlocal conservation laws and a nonlocal form of Noether’s theorem, (c) Implementation of the ghost calculus in standard computer algebra packages.

For the entire collection see [Zbl 1050.35003].

For the entire collection see [Zbl 1050.35003].

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

35R10 | Functional partial differential equations |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

76M60 | Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics |